- individual particles have little movement regardless of the distance of wave
- **transfer of energy without net transfer of matter**
-**Nodes** - point of no motion (fixed point on graph)
+**Nodes** - point of no motion (fixed point on graph)
**Antinodes** - point of maximum motion (peaks)
**Crests** (peaks) & **troughs** (azimuths)
### Measuring mechanical waves
-**Amplitude $A$** - max displacement from rest position (0)
-**Wavelength $\lambda$** - distance between two points of same y-value (points are in phase)
+**Amplitude $A$** - max displacement from rest position (0)
+**Wavelength $\lambda$** - distance between two points of same y-value (points are in phase)
**Frequency $f$** - number of cycles (wavelengths) per second
-$T={1 \over f}\quad$(period: time for one cycle)
+$T={1 \over f}\quad$(period: time for one cycle)
$v=f \lambda \quad$(speed: displacement per second)
### Doppler effect
## Interference patterns
When a medium changes character:
+
- some energy is *reflected*
- some energy is *absorbed* by new medium
- some energy is *transmitted*
-**Superposition** - stimuli add together at a given point (vector addition)
+**Superposition** - stimuli add together at a given point (vector addition)
**Standing wave** - constructive interference at resonant frequency
### Reflection
Angle of incidence $\theta_i =$ angle of reflection $\theta_r$
+
- Normal: $\perp$ to wall
- Incident wave front: $\perp$ to incident ray
-- Incident ray: $
+- Incident ray: $\theta_i$
#### Transverse
- sign of reflected transverse wave is inverted when endpoint is fixed in y-axis (equivalent to $180^\circ=\pi={\lambda \over 2}$ phase change)
**Overtone** - a multiple of the fundamental harmonic which produces the same no. of wavelengths at a different frequency (due to constructive interference)
#### Wave has antinodes at both ends:
-$\lambda = {{2l} \div n}\quad$ (wavelength for $n^{th}$ harmonic)
+$\lambda = {{2l} \div n}\quad$ (wavelength for $n^{th}$ harmonic)
$f = {nv \div 2l}\quad$ (frequency for $n_{th}$ harmonic at length $l$ and speed $v$)
#### Wave has antinode at one end:
-$\lambda = {{4l} \div n}\quad$ (wavelength for $n^{th}$ harmonic)
+$\lambda = {{4l} \div n}\quad$ (wavelength for $n^{th}$ harmonic)
$f = {nv \div 4l}\quad$ (frequency for $n_{th}$ harmonic at length $l$ and speed $v$)
## Light
### Double Slit
-**(a) Double slit as theorised by particle model** - "streams" of photons are concentrated in bright spots
+**(a) Double slit as theorised by particle model** - "streams" of photons are concentrated in bright spots
**(b) Double slit as theorised by wave model** - waves disperse onto screen (overlapping)
Young's double slit experiment supports wave model:
Path difference $pd = |S_1P-S_2P|$ for point $p$ on screen
-Constructive interference when $pd = n\lambda$ where $n \in [0, 1, 2, ...]$
+Constructive interference when $pd = n\lambda$ where $n \in [0, 1, 2, ...]$
Destructive interference when $pd = (n-{1 \over 2})\lambda$ where $n \in [1, 2, 3, ...]$
Fringe separation:
$$\Delta x = {{\lambda l }\over d}$$
where
-$\Delta x$ is distance between fringes
-$l$ is distance from slits to screen
+$\Delta x$ is distance between fringes
+$l$ is distance from slits to screen
$d$ is separation between sluts ($=S_1-S_2$)
## Electromagnetic waves
$L^+$ - limit from above
-$\lim_{x \to a} f(x)$ - limit of a point
+$\lim_{x \to a} f(x)$ - limit of a point
- Limit exists if $L^-=L^+$
- If limit exists, point does not.
## First principles derivative
-$$\lim_{\delta x \rightarrow 0}{\delta y \over \delta x}={dy \over dx} = f^\prime(x)$$
+$$f^\prime(x) = \lim_{\delta x \rightarrow 0}{\delta y \over \delta x}={dy \over dx}$$
$$m_{\operatorname{tangent}}=\lim_{h \rightarrow 0}f^\prime(x)$$
$$m_{\operatorname{chord PQ}}=f^\prime(x)$$
first principles derivative:
-$${m_{\operatorname{tangent at P}} =\lim_{h \rigzhtarrow 0}}{{f(x+h)-f(x)}\over h}$$
+$${m_{\operatorname{tangent at P}} =\lim_{h \rightarrow 0}}{{f(x+h)-f(x)}\over h}$$
+## Gradient at a point
+Given point $P(a, b)$ and function $f(x)$, the gradient is $f^\prime(a)$
+## Derivatives of $x^n$
+
+For $f: \mathbb{R} \rightarrow \mathbb{R}$ where $f(x)=x^n, x \in \mathbb{N}$
+
+Derivative is $f^\prime(x) = nx^{n-1}$
+
+If $x=$ constant, derivative is $0$
+
+If $f(x)={1 \over x}=x^{-1}, \quad f^\prime(x)=-1x^{-2}={-1 \over x^2}$
+
+If $f(x)=^5\sqrt{x}=x^{1 \over 5}, \quad f^\prime(x)={1 \over 5}x^{-4/5}={1 \over 5 \times ^5\sqrt{x^4}}$
+
+If $f(x)=(x-b)^2, \quad f^\prime(x)=2(x-b)$
+
+$$f^\prime(x)=\lim_{h \rightarrow 0}{{f(x+h)-f(x)} \over h}$$
+$$=\lim_{h \rightarrow 0}
+
## Euler's number as a limit
$$\lim_{h \rightarrow 0} {{e^h-1} \over h}=1$$